Ginzburg-Landau form

Here M is a mobility coefficient, which is assumed to be constant and r/(r.t) is the random thermal noise term, which for a system in equilibrium at temperature T satisfies the fluctuation-dissipation theorem. The free energy functional is taken to be of a Ginzburg-Landau form. In the notation of Qi and Wang (1996,1997) it is given by... 

Within this continuum approach Cahn and Hilliard [481 have studied the universal properties of interfaces. While their elegant scheme is applicable to arbitrary free-energy functionals with a square gradient form we illustrate it here for the important special case of the Ginzburg-Landau form. For an ideally planar interface the profile depends only on the distance z from the interfacial plane. In mean field approximation, the profile m(z) minimizes the free-energy functional (B3.6.11). This yields the Euler-Lagrange equation... 

In the previous sections, we briefly introduced a number of different specific models for crystal growth. In this section we will make some further simplifications to treat some generic behavior of growth problems in the simplest possible form. This usually leads to some nonlinear partial differential equations, known under names like Burgers, Kardar-Parisi-Zhang (KPZ), Kuramoto-Sivashinsky, Edwards-Anderson, complex Ginzburg-Landau equation and others. 

Fig. 2.46 Cell-dynamical simulation of a symmetric block copolymer in two dimensions (64 X 64 lattice) (Hamley 1997).This structure forms from an initially homogeneous state via time-dependent Ginzburg-Landau kinetics at a temperature below the ODT.

Fig. 2.53 Computer simulation results, using lime-dependent Ginzburg-Landau dynamics, of a lattice model of an asymmetric copolymer forming a hex phase subject to a step-shear along the horizontal axis (Ohta et al. 1993), The evolution of the domain pattern after the application of the step-shear is shown, (a) t = 1 (the pattern immediately after the shear is applied) (b) t = 5000 (c) t = 10000 (d) t = 15 000. The time-scale corresponds to the characteristic time for motion of an individual chain, t = R M. 

In some cases, one is interested in the structures of complex fluids only at the continuum level, and the detailed molecular structure is not important. For example, long polymer molecules, especially block copolymers, can form phases whose microstructure has length scales ranging from nanometers almost up to microns. Computer simulations of such structures at the level of atoms is not feasible. However, composition field equations can be written that account for the dynamics of some slow variable such as 0 (x), the concentration of one species in a binary polymer blend, or of one block of a diblock copolymer. If an expression for the free energy / of the mixture exists, then a Ginzburg-Landau type of equation can sometimes be written for the time evolution of the variable 0 with or without flow. An example of such an equation is (Ohta et al. 1990 Tanaka 1994 Kodama and Doi 1996)... 

The resulting free energy expansion is found to have the same symmetry (Alexander, 1975) as that of the 3-state Potts model, cf. eqs. (22)-(25). The general form of the Ginzburg-Landau-Wilson -Hamiltonian F ( )) then is (see also Straley and Fisher, 1973 Stephanov and Tsypin, 1991)... 

This makes possible the introduction of a Ginzburg-Landau (GL) formalism. We use the simplest possible form of GL free-ensrgy functional... 

With the form of free energy functional prescribed in equation (A3.3.52). equation (A3.3.43) and equation (A3.3. 48) respectively define the problem of kinetics in models A and B. The Langevin equation for model A is also referred to as the time-dependent Ginzburg-Landau equation (if the noise term is ignored) the model B equation is often referred to as the Cahn-Hilliard-Cook equation, and as the Cahn-Hilliard equation in the absence of the noise term. 

This form is called a Ginzburg-Landau expansion. The first term f(m) corresponds to the free energy of a homogeneous (bulk-like) system and determines the phase behaviour. For t > 0 the function/exhibits two minima at nj = /. This value corresponds to the composition difference of the two coexisting phases. The second contribution specifies the cost of an inhomogeneous order parameter profile. / sets the t5 ical length scale. 

Figure 21. Numerical solution of the complex Ginzburg-Landau equation (159) in the form of a spiral wave.

The scattering intensity in bulk contrast can be calculated easily in the Ornstein-Zernike approximation for all lattice [15, 90-92] and Ginzburg-Landau models. In the limit of wave vector q < q — n/a, one obtains in all cases the Teubner-Strey form... 

Finally, a quench into the one-phase region of the microemulsion has been investigated. An analysis based on a Ginzburg-Landau model for a single, conserved order parameter predicts [160] that the equal-time structure factor, Eq. (65), approaches its equilibrium form S(k) algebraically for long times t. 

Next came the likewise phenomenological Ginzburg-Landau theory of superconductivity, based on the Landau theory of a second-order phase transition (see also Appendix B) that predicted the coherence length and penetration depth as two characteristic parameters of a superconductor (Ginzburg and Landau, 1950). Based on this theory, Abrikosov derived the notion that the magnetic field penetrates type II superconductors in quantized flux tubes, commonly in the form of a hexagonal network (Abrikosov, 1957). The existence of this vortex lattice was... 

In Appendix A we described the numerical implementation of the Gaussian chain density functions that retains the bijectivity between the density fields and the external potential fields. The intrinsic chemical potentials /r/= SFj p that act as thermodynamic driving forces in the Ginzburg-Landau equations describing the dynamics, are functionals of the external potentials and the density fields. Together, the Gaussian chain density functional and the partial differential equations, describing the dynamics of the system, form a closed set. 

Small-amplitude oscillations near the Hopf bifurcation point are generally governed by a simple evolution equation. If such oscillators form a field through diffusion-coupling, the governing equation is a simple partial differential equation called the Ginzburg-Landau equation. 

It is sometimes more convenient to work with a further rescaled form of the Ginzburg-Landau equation. A suitable scale transformation would be... 

Since no confusion is expected, we shall express the scaled equation in terms of the more natural notations t and r in place of t and s. Then, under the condition (2.4.12), the Ginzburg-Landau equation above criticality reduces to the form... 

It is important to realize that the Ginzburg-Landau equation is not a special case of (2.4.17). In fact, (2.4.13) may at best be written in the form... 

It would be instructive here to illustrate the theory presented above with a simple reaction-diffusion model. A suitable model would be the Ginzburg-Landau equation in the form of (2.4.13). As noted in Sect. 2.4, it is expressed as a two-component reaction-diffusion system, although the diffusion matrix D then involves an antisymmetric part ... 

---